My understanding of ITIL tells me that these type of things should be handled as problems under exact customer service. My boss on the other hand thinks that for these type of situations should exist separate technical services like billing or customer care. Do you have any thought on how to handle this? I would be also interested in hearing how you implemented this part of service desk with OTRS.

It really depends on how you want to report on them. If you have a set of predefined scopes (ie, how your boss wants to do it), reporting is easier. It's less cluttered for the people entering to use your way. Both work, but very few people understand their business well enough to really enumerate it as separate services, with all the documentation for a service. We did a hybrid approach: do the service definition for common problems, and handle everything else as a problem. We then periodically go back, look at the "problem" entries, create new service definitions for them, and reclassify the "problem" entries accordingly. We did it this way to avoid having to think of every possible thing at the start, and to reinforce the culture of continuous improvement over time.